Branching Bezier Curves
If you want it to branch you need to use a mesh, not curves.

If you want your mesh to have an all-quad topology, here is the simplest way:



You can then further subdivide the edge rings to add detail and round off the roots more:



And with some additional fine-tuning of vertices and edge loops you can get something like this:



If you really want to use Bezier curves I recommend using the Sapling Add-on, which is packaged with Blender and just needs to be enabled from Preferences. It's ideal for making trees and other branching foliage. Although for a character such as the one you posted, I would model it as a mesh.